Dominant height growth and dynamic site index models for Crimean pine in Kastamonu-Taşköprü region of Turkey

Abstract

Some dynamic site index models based on the generalized algebraic difference approach (GADA) were fitted for Crimean pine (Pinus nigra J.F. Arnold subsp. pallasiana (Lamb.) Holmboe) stands in Taşköprü, Turkey. Data were obtained from 132 dominant trees representing the wide range of site quality in the region. Nonlinear regression analysis and a second-order continuous-time autoregressive error structure were applied. After autoregressive modeling, the fitted models were evaluated both statistically and graphically. The best results were obtained with the dynamic site index model derived from the Bertalanffy- Richards base equation, accounting for about the 99% of the total variance in height-age relationships in dominant trees, with an Akaike information criterion (AIC) value of 119.55 and root mean square error (RMSE) of 0.5446. The selected base-age invariant dynamic site index curves provided the polymorphism with multiple asymptotes and other realistic height growth patterns.

Full text

ARTICLE
Dominant height growth and dynamic site index models for
Crimean pine in the Kastamonu–Tas¸köprü region of Turkey
Mehmet Seki and Oytun Emre Sakici
Abstract: Some dynamic site index models based on the generalized algebraic difference approach (GADA) were fitted for
Crimean pine (Pinus nigra J.F. Arnold subsp. pallasiana (Lamb.) Holmboe) stands in Tas¸köprü, Turkey. Data were obtained from
132 dominant trees representing the wide range of site quality in the region. Nonlinear regression analysis and a second-order
continuous-time autoregressive error structure were applied. After autoregressive modeling, the fitted models were evaluated
both statistically and graphically. The best results were obtained with the dynamic site index model derived from the Bertalanffy–
Richards base equation, accounting for about the 99% of the total variance in height–age relationships in dominant trees, with an
Akaike information criterion (AIC) value of 119.55 and root mean square error (RMSE) of 0.5446. The selected base-age invariant
dynamic site index curves provided the polymorphism with multiple asymptotes and other realistic height growth patterns.
Key words: site quality, generalized algebraic difference approach, base-age invariant equations, dynamic equations, Crimean pine.
Résumé : Quelques modèles dynamiques d’indice de qualité de station ont été ajustés a
`l’aide de l’approche de la différence
algébrique généralisée (GADA) pour des peuplements de pin de Crimée (Pinus nigra J.F. Arnold ssp. pallasiana (Lamb.) Holmboe) de
Tas¸köprü, en Turquie. Les données ont été obtenues de 132 arbres dominants couvrant une grande étendue de qualité de station
de la région. Nous avons appliqué une analyse de régression non linéaire et une structure d’erreur autorégressive a
`série
temporelle continue de deuxième degré. Après la modélisation autorégressive, les modèles ajustés ont été évalués de façons
statistique et graphique. Les meilleurs résultats ont été obtenus a
`partir du modèle dynamique d’indice de qualité de station
découlant de l’équation de base de Bertalanffy et Richards. Ce modèle expliquait environ 99 % de la variance totale des relations
entre la hauteur et l’âge des arbres dominants, avec une valeur du critère d’information d’Akaike (AIC) de 119,55 et une erreur
quadratique moyenne (EQM) de 0,5446. Les courbes sélectionnées d’indice de qualité de station dynamiques et indépendantes de
l’âge de référence affichaient un polymorphisme avec plusieurs asymptotes et d’autres caractéristiques réalistes de la croissance
en hauteur. [Traduit par la Rédaction]
Mots-clés : qualité de station, approche de la différence algébrique généralisée, équations indépendantes de l’âge de référence,
équations dynamiques, pin de Crimée.
1. Introduction
Pinus nigra J.F. Arnold is a widespread and valuable tree species
in Turkey, with total forest area of about 4.7 million ha (General
Directorate of Forestry 2013). This species grows naturally in
southern Europe, the Balkans, and western Asia in both pure and
mixed stands. It has high quality wood and shallow roots, prefers
arid, rocky and poor soils, and can live for centuries. Pinus nigra
subsp. pallasiana (Lamb.) Holmboe (Crimean pine), one of five sub-
species of P. nigra, grows naturally in Turkey and is widely distrib-
uted in the western Black Sea and Anatolian regions and on the
northern side of the Taurus Mountains (Akman et al. 2003;
Mamıkog˘lu 2007).
Estimation of forest productivity is very crucial for both forest
management and ecological studies in terms of accurate assess-
ment of site conditions. Site productivity is influenced by site-
related and climatic factors such as temperature, light intensity,
moisture, nutrients, and soil properties (Wang and Klinka 1996).
Estimation of site quality generally depends on the variable height,
as there is a direct correlation between site quality and height
growth. As mean height of the stand can be affected by silvicul-
tural treatments, dominant stand height is used to quantify site
productivity (Clutter et al. 1983;Monserud 1984). Site index (SI),
the average height of dominant trees at a specified base age, is the
most common variable used to evaluate productivity of forest
areas (Carmean 1972;Clutter et al. 1983;Carmean and Lenthall
1989;Payandeh and Wang 1994;Barrio-Anta and Diéguez-Aranda
2005).
From past to present, site index curves drawn by hand, empir-
ical models, and biologically based functions have been used to
model site index. With the development of technology and calcu-
lation techniques and increased information about height growth
processes, site index modeling has become an important area of
forest research. Initially, the main purpose of site index modeling
was to define probable productivity of forest areas when forest
researchers were primarily interested in growth and yield. Re-
cently, with the increase in the importance of the concepts of
multipurpose, protection, and biomass, more advanced models
are needed to understand the processes of these concepts in forest
ecosystem. Site index and height growth models are the types of
models that need to be improved in this aspect (Bravo-Oviedo et al.
2007).
There are two model forms used for site index modeling: static
and dynamic site index equations. The general forms of static and
dynamic site index equations are, respectively, Y=f(t,S), where
variable Yis the predicted dominant height at age tand S is the
Received 29 March 2017. Accepted 31 July 2017.
M. Seki and O.E. Sakici. Kastamonu University, Faculty of Forestry, Department of Forest Engineering, Kastamonu, Turkey.
Corresponding author: Mehmet Seki (email: mseki@kastamonu.edu.tr).
Copyright remains with the author(s) or their institution(s). Permission for reuse (free in most cases) can be obtained from RightsLink.
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specified base age (Diéguez-Aranda et al. 2006), and Y=f(t,t
0
,Y
0
),
where Yis the predicted dominant height at age tand Y
0
is the
reference variable described as the value of the equation at age t
0
(Cieszewski and Bailey 2000). The first step in fitting static site
index equations is determining the base age (Stage 1963;Curtis
et al. 1974;Monserud 1984;Bruchwald 1988). Selecting the base
age is crucial for the fixed base-age variant site index models
because the specific base age affects the estimations and domi-
nant height estimations are dependent on base age (Bailey and
Cieszewski 2000). The importance of dynamic site index models
has been underlined in numerous investigations in terms of their
realistic growth patterns, base-age invariant properties, and their
ability to provide the same estimates for any base age (Cieszewski
and Bailey 2000;Cieszewski 2001,2002,2003;Diéguez-Aranda
et al. 2005;Ercanlı et al. 2014;Tewari et al. 2014).
Bailey and Clutter (1974) produced a procedure known as the
algebraic difference approach (ADA) in which the dynamic site
index predictions are not affected by the specific base age. These
dynamic site index models are base-age invariant and indepen-
dent of base-age selection, making one parameter site specific.
They have biologically realistic growth characteristics such as
variable asymptotes and polymorphism in curve shapes (Cieszewski
and Bailey 2000;Diéguez-Aranda et al. 2006;Tewari et al. 2014).
However, site index models produced by the ADA are either ana-
morphic or have single asymptotes, while one of the biologically
realistic characteristics of dominant height growth is polymor-
phism with variable asymptotes (Cieszewski 2002).
The ADA methodology was generalized by Cieszewski and
Bailey (2000) and entered the literature as the generalized alge-
braic difference approach (GADA). The new methodology allows
more than one parameter to be site specific and can produce both
polymorphic and multiple asymptotic site index curves and be
base-age invariant at the same time. The new models derived with
GADA methodology provide the best predictions in terms of de-
sirable height growth patterns compared with previous site index
modeling approaches. Thus, the GADA methodology has become
the most common technique for site index modeling in the liter-
ature (Diéguez-Aranda et al. 2006).
In the forests of Turkey, forest researchers and forest managers
still use site index curves developed by Kalıpsız (1963) to estimate
the site index and determine the productivity of Crimean pine
stands. The new dynamic site index curves are needed for more
qualified estimates of stand productivity for these stands, with
more realistic growth features. The objectives of this study are
(i) to develop dynamic site index models derived by the GADA
methodology and (ii) to correct inherent autocorrelation of the
data used for pure Crimean pine stands in the Tas¸köprü region of
northwestern Turkey.
2. Material and methods
2.1. Study area
This study investigated the pure Crimean pine stands of Tas¸köprü
Forest Enterprise, which is 42 km from Kastamonu city in Turkey
(Fig. 1). The total area of the studied region is 176.648 ha, and
forests cover 64% of the region.
Elevation of the study area varies from 812 to 1524 m above sea
level, with an average of 1169 m, and slope ranges between 2% and
41%. The study area has a mild oceanic climate and cool to cold
rainy winters and receives high amounts of precipitation through-
out the year. When the history of the sampled stands was investi-
gated, there was no evidence of forest fire, insect, or other harmful
effects.
All of the sampled area is naturally regenerated and pure Crimean
pine stands representing a wide range of ages, densities, and sites
with diverse ecological properties differently impacting the growth
rate. With these properties, the sampled area is very important both
ecologically and economically.
2.2. Data
For this study, 132 dominant trees were selected for stem anal-
ysis. The sample trees were obtained from sample plots randomly
selected to represent the available range of sites, densities, and
ages in the studied area. All of the sample trees were felled and
cross-sectioned at stump height (0.30 m), breast height (1.30 m),
2.30 m, and intervals every 1–2 m along the tree stem thereafter.
The number of annual rings was counted at each section, and ages
at different heights were calculated. To determine ages of sample
trees, three years (average age of Crimean pine at 0.30 m in this
region) were added to the number of rings counted at stump
height. Because cross-section lengths do not coincide with peri-
odic height growth, adjusting height–age data from stem analysis
Fig. 1. Geographic location of the study area.
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has been highly recommended (Calama et al. 2003;Diéguez-Aranda
et al. 2005;Ercanlı et al. 2014;López-Sánchez et al. 2015). The iter-
ative screening and structure analysis (ISSA) method using the
second differences of ring counts to obtain smoother height–age
curves presented by Fabbio et al. (1994) was used to calculate the
height for each tree age, and a total of 1350 height–age pairs from
the analysis of 132 stem was used for the fitting of the site index
equations (Fig. 2). The statistics for the sample trees, including
minimum, maximum, mean, and standard deviation (SD) values
of age (t), height (h), and diameter at breast height (dbh), are sum-
marized in Table 1.
2.3. Candidate functions
To examine height growth of dominant trees, various growth
models are present in the literature. Hossfeld (Hossfeld 1882),
Lundqvist–Korf (Korf 1939;Lundqvist 1957), Bertalanffy–Richards
(Bertalanffy 1949,1957;Richards 1959), and King–Prodan (Prodan
1951;King 1966) models are the most common and appropriate
growth models used for the height growth modeling. In this
study, we used five GADA models (Table 2) derived from these base
model forms by Cieszewski (2002,2004) and Krumland and Eng
(2005). These GADA models are the polymorphic versions of the
base growth models and have the base form of h=f(h
0
,t
0
,t,b
1
,
b
2
,…,b
n
)(Krumland and Eng 2005).
Dynamic site index models derived with GADA methodology
have one or two site-specific parameters related to site quality
(Diéguez-Aranda et al. 2006), while the features of polymorphism
and multiple asymptotes can be provided by the GADA models
with two site-specific parameters. These features have great im-
portance for site index models (Cieszewski 2002). In this study,
five GADA models with two site-specific parameters were selected
as candidate functions. The most important characteristic of the
candidate functions is the ability to provide curves featuring poly-
morphism and multiple asymptotes.
2.4. Modeling approach
Nonlinear regression analysis was used for fitting the five dy-
namic site index models derived with the GADA. In this study, for
the nonlinear regression analysis executed using the Marquardt
algorithm, which is one of the most useful methods for im-
mensely correlated parameter estimates (Fang and Bailey 1998;
Kelley 1999;Martín-Benito et al. 2008), the SAS/ETS PROC MODEL
procedure in SAS software (SAS Institute Inc. 2004) was used.
It is assumed that error terms are unrelated random variables in
regression analysis. However, in time series data, these terms are
related to each other and are called “correlated errors”. For the
polymorphic site index modeling approach, stem analysis data
that have the time series feature and autocorrelation problem are
used. Time series data structure such as data obtained from stem
analysis has an autocorrelation problem that could be modeled by
the autoregressive modeling approach. Therefore, a continuous-
time autoregressive error structure is highly recommended by
forest researchers (Monserud 1984;Grégoire et al. 1995;Parresol
and Vissage 1998). The autocorrelation problem is often seen in
equations for a single time series, and unstable error variances
might be seen in a cross-sectional regression. If time series and
cross-sectional data are combined, these problems could be seen
at the same time (Dielman 1989;Diéguez-Aranda et al. 2005). The
shape of site index curves should not be changed after correction of
autocorrelation compared with the uncorrected fitting (Cieszewski
2003). In this study, autocorrelation was modeled by the AR(2) error
structure; expansion of the error term and the model structure is
given as follows (Gea-Izquierdo et al. 2008):
Ei⫽p1Ei⫺1⫹p2Ei⫺2
where Eiis the residual for observation iand p
1
and p
2
are param-
eters for autocorrelation.
For parameter estimation, the SAS/ETS MODEL procedure in
SAS software (SAS Institute Inc. 2004) was used. After autoregres-
sive modeling, the Durbin–Watson statistic (d) was calculated to
detect if autocorrelation in the residuals was removed or not, and
graphs representing residuals versus lag residuals from previous
observations were examined visually. The possible problem of
unequal error variance (heteroscedasticity) was also investigated
by visual comparison of residual plots against predicted heights.
The Durbin–Watson statistic takes a value between 0 and 4: d<2
means positive correlation, d> 2 means negative correlation, and
d≈ 2 means there is no autocorrelation (Fox 1997). The Durbin–
Watson statistic is expressed as follows:
d⫽兺i⫽2
n(ei⫺ei⫺1)2
兺i⫽2
nei
2
where e
i
and e
i–1
are the error terms for iand i–1, and nis the
number of data.
2.5. Model selection criteria
A three-step procedure was performed for evaluation and selec-
tion of the GADA models (Palahí et al. 2004;Adame et al. 2006;
Martín-Benito et al. 2008;Ercanlı et al. 2014). This procedure is
based on both graphical and numerical analyses of the residuals.
In the first step, three statistics were used: root mean square error
(RMSE), adjusted coefficient of determination (Radj
2), and the
Akaike information criterion (AIC). The statistics were calculated
as follows:
RMSE ⫽冑兺i⫽1
n(hi⫺h
ˆi)2
n⫺p
Radj
2⫽兺i⫽1
n(hi⫺h
ˆi)2
兺i⫽1
n(hi⫺h
¯
i)2
(n⫺1)
(n⫺p)
Fig. 2. The profile plots of 132 stem analyses.
0
5
10
15
20
25
30
35
020
40 60 80 100 120 140 160 180
Table 1. Summary statistics of the sample trees.
Variable nMinimum Maximum Mean SD
Height (h) (m) 132 8.2 34.4 17.3 5.2
Age (t) (years) 132 27 170 85 31
Diameter at breast
height (dbh) (cm)
132 11.5 52.6 27.6 7.1
Seki and Sakici 1443
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AIC ⫽2k⫺2log L
where h
i
,h
ˆi, and h
¯
iare the observed, estimated, and mean values of
the dominant heights, respectively; nis the number of data used
for fitting; pis the number of parameters; kis the number of
estimable parameters; and Lis the likelihood at its maximum
point of the model estimated.
In the second step, characterizations of model residuals were
examined graphically. Finally, growth patterns and desirable bio-
logical characteristics of the models fitted were taken into ac-
count. These desirable attributes were polymorphism, multiple
asymptotes, origin point at time 0, sigmoid growth pattern, and
other theoretical facts about height growth.
3. Results
Firstly, nonlinear least squares without expanding the error
terms accounting for autocorrelation was used for fitting the
dynamic site index models (Table 3). As seen in this table, the
Durbin–Watson statistics (d) are much lower than 2 (between
0.2999 and 0.3773), meaning that positive correlation appears
in the residuals. After correcting for autocorrelation, the pre-
dictions of models’ parameters and their goodness-of-fit statis-
tics were determined (Table 4). The Durbin–Watson statistics (d)
approached the value of approximately 2, except model 1 (d=
2.1155, significant at p< 0.05), which still had autocorrelation
(Table 4). Other than the autocorrelation problem, which was
removed from the estimates, the statistical properties of the mod-
els were improved by autoregressive modeling as seen in Tables 3
and 4.
All of the model parameters were found to be significant at
p< 0.05, excluding parameter b
2
of model 5. Model 5 with the
nonsignificant parameter and model 1 with high Durbin–Watson
(d) and AIC values were eliminated in this step. As seen from
Table 4, the other three GADA models (models 2 to 4) showed
high-quality predictions with high Radj
2and low AIC and RMSE
values. More than 99% of the total variance was explained by the
three models, and the models showed similar fitting values.
Patterns of residuals for predicted heights by models 2, 3, and 4
are presented graphically, and all of the models ensured indis-
criminate patterns of residuals around zero and no apparent
trends with homogenous variance until high values of predicted
heights. As well, the plots of observed and predicted heights were
drawn and model fits seemed good (Fig. 3). As seen in Fig. 3, the
residuals are much lower for the high tree heights than for other
height classes. The main reasons for this changing variance are
(i) the number of sample trees taller than 30 m is less than the
Table 2. Models used to fit the site index curves.
Base form of model
Site-specific
parameters Solution for variable XGADA formulation
Hossfeld: h⫽a
1⫹bt⫺c
a=b
1
+X
b=b
2
/X
X0⫽1
2关h0⫺b1⫺
兹
共h0⫺b1兲2⫺4b2h0t0
⫺b3兴h⫽b1⫹X0
1⫹b2/X0t⫺b3
Cieszewski (2002) (model 1)
Lundqvist–Korf:
h=aexp(–bt
–c
)
a= exp(x)
b⫽b1⫹b2
X
X0⫽1
2共b1t0
⫺b3⫹ln h0⫹L0兲
L0⫽
兹
共b1t0
⫺b3⫹ln h0兲2⫹4b2t0
⫺b3
h⫽exp共X0兲exp
冋
⫺
冉
b1⫹b2
X0
冊
t⫺b3
册
Cieszewski (2004) (model 2)
Bertalanffy–Richards:
h=a[1 – exp(–bt)]
c
a= exp(X)
c=b
2
+b
3
/X
X0⫽1
2关共ln h0⫺b2L0兲⫹
兹
共ln h0⫺b2L0兲2⫺4b3L0兴
L0⫽ln关1⫺exp共⫺b1t0兲兴 h⫽h0
冋
1⫺exp共⫺b1t兲
1⫺exp共⫺b1t0兲
册
冉
b2⫹
b3
X0
冊
Cieszewski (2004) (model 3)
a= exp(X)
c=b
2
+1/X
X0⫽1
2关共ln h0⫺b2L0兲⫹
兹
共ln h0⫺b2L0兲2⫺4L0兴
L0⫽ln关1⫺exp共⫺b1t0兲兴
h⫽exp共X0兲关1⫺exp共⫺b1t兲兴共b2⫹1/X0兲
Cieszewski (2004) (model 4)
King–Prodan:
h⫽ta
b⫹cta
b=b
2
+b
3
/X
c=XX0⫽
冉
t0b1
h0
冊
⫺b2
b3⫹t0b1
h⫽tb1
b2⫹b3X0⫹X0tb1
Krumland and Eng (2005) (model 5)
Note: a,b, and care parameters in base growth model forms; b
1
,b
2
, and b
3
are parameters in dynamic site index models; hand h
0
are heights (m) at age
tand t
0
(years), respectevely; X
0
is the solution of the Xparameter relating site.
Table 3. Parameter estimates and fit statistics for models tested.
Model Radj
2AIC RMSE dParameter Estimate
Standard
error tvalue
Approx.
pvalue
1 0.8653 2464.78 2.4887 0.3773 b
1
34.61231 0.4627 67.17 <0.0001
b
2
–112222 20 698.1 –4.69 <0.0001
b
3
2.561753 0.0499 43.36 <0.0001
2 0.9752 177.99 1.0670 0.2999 b
1
16.3842 1.0730 15.27 <0.0001
b
2
–20.1342 5.2298 –3.85 0.0001
b
3
0.425651 0.0160 26.66 <0.0001
3 0.9769 82.04 1.0296 0.3147 b
1
0.014116 0.0004 32.25 <0.0001
b
2
1.907675 0.1122 17.01 <0.0001
b
3
–1.38351 0.3965 –3.49 0.0005
4 0.9765 107.65 1.0400 0.3113 b
1
0.014671 0.0004 35.10 <0.0001
b
2
1.258183 0.0229 54.90 <0.0001
5 0.9767 97.69 1.0357 0.3100 b
1
1.472684 0.0189 78.11 <0.0001
b
2
1.769255 1.3547 1.31 0.1918ns
b
3
690.1876 70.6661 9.77 <0.0001
Note: Radj
2, adjusted coefficient of determination; AIC, Akaike information criterion; RMSE, root mean square error; d, Durbin–
Watson statistics; ns, nonsignificant at the 0.05 level.
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Table 4. Parameter estimates and fit statistics after autoregressive modeling.
Model Radj
2AIC RMSE dParameter Estimate
Standard
error tvalue
Approx.
pvalue
1 0.9646 3271.40 1.2764 2.1155 b
1
33.73361 0.4446 75.87 <0.0001
b
2
–92501.1 19 438.8 –4.76 <0.0001
b
3
2.744842 0.0623 44.05 <0.0001
2 0.9934 178.83 0.5503 2.0518 b
1
16.82822 1.6074 10.47 <0.0001
b
2
–23.48 7.8039 –3.01 0.0027
b
3
0.415299 0.0195 21.25 <0.0001
3 0.9936 91.53 0.5422 2.0596 b
1
0.0142 0.0006 22.82 <0.0001
b
2
1.953322 0.1847 10.57 <0.0001
b
3
–1.61622 0.6406 –2.52 0.0118
4 0.9936 119.55 0.5446 2.0581 b
1
0.014686 0.0006 24.37 <0.0001
b
2
1.227079 0.0328 37.45 <0.0001
5 0.9936 111.28 0.5408 2.0602 b
1
1.46529 0.0265 55.30 <0.0001
b
2
0.662246 2.2663 0.29 0.7702ns
b
3
678.2673 108.01 6.28 <0.0001
Note: Radj
2, adjusted coefficient of determination; AIC, Akaike information criterion; RMSE, root mean square error; d, Durbin–
Watson statistics; ns, nonsignificant at the 0.05 level.
Fig. 3. Plots of observed and predicted dominant heights (left) and residuals vs. predicted dominant heights (right) for models 2, 3, and 4.
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35
Observed dominant height (m)
Predicted dominant height (m)
-6
-4
-2
0
2
4
6
0 5 10 15 20 25 30 35
Residuals (m)
Predicted dominant height (m)
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35
Observed dominant height (m)
Predicted dominant height (m)
-6
-4
-2
0
2
4
6
0 5 10 15 20 25 30 35
Residuals (m)
Predicted dominant height (m)
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35
Observed dominant height (m)
Predicted dominant height (m)
-6
-4
-2
0
2
4
6
0 5 10 15 20 25 30 35
Residuals (m)
Predicted dominant height (m)
Model 2 Model 3 Model 4
Seki and Sakici 1445
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number of sample trees in other height classes, and (ii) sample
trees were obtained from plots that varied ecologically. As men-
tioned in section 2.2 (data), we tried to take sample plots from a
wide range of ages, densities, and sites. However, in the sampled
area, only a few Crimean pine stands that included trees taller
than 30 m were found and sampled. As expected, height classes up
to 30 m showed homogenous variance of residuals and this rate
decreased after predictions from trees taller than 30 m.
In Fig. 4 (first and second columns), after correction for autocor-
relation, there was no strong temporal autocorrelation seen be-
tween residuals when we examined the AR(2) residuals versus
age-lag1 and age-lag2 residuals for all models. Both Durbin–
Watson tests (not significant, p> 0.05) and age-lag graphs showed
that strong temporal autocorrelation is not seen in residuals. Also,
as shown in Fig. 4 (third column), no trend was seen in residuals as
a function of height-lag1 residuals for all models. This means that
sample trees and plots were selected randomly and independently
and there is no evidence of subjective selection. These results
were similar to those of Cieszewski (2003) and Diéguez-Aranda
et al. (2005).
In the last step, desirable biological presumptions of dominant
height growth were examined for models 2, 3, and 4. When we
examined the behaviors of the site index curves for these three
models, the models provided almost all of the desirable biological
presumptions of site index models such as polymorphism, multi-
ple asymptotes, origin point at time 0, and sigmoid growth pat-
tern. However, only model 4 provided the value of mean annual
increment (MAI) increase, the age of reaching maximum MAI ex-
pressed as inflection point decrease with site index values, for
more productive sites.
For model comparisons for biological presumptions, selection
of base age is an important issue. When we examined the relative
errors (RE%) of different base ages with their corresponding ob-
served ages, base ages of 90 and 100 years provided the best pre-
dictions (Fig. 5). In spite of the RE% for base age 90 years being
lower, the number of observations at this age is lower than at base
age 100 years. Also, one of indications of Goelz and Burk (1992) is
that base age should be less than or equal to the youngest rotation
in typically managed forests. The rotation age of Crimean pine in
the studied area is 120 years for moderate sites, and the base age of
100 years for Crimean pine is generally used in Turkey; therefore,
100 years was selected as the base age in this study.
As mentioned above, the biological presumptions checked are
polymorphism, multiple asymptotes, and sigmoid growth pat-
tern. For this purpose, site index curves with site indexes (SIs) of
10, 15, 20, 25, and 30 m at the reference age of 100 years for
models 2, 3, and 4 are given in Fig. 6. When the graphs are exam-
ined, it can be seen that the site index curves provide the desirable
Fig. 4. Residuals vs. age-lag1, age-lag2, and height-lag1 residuals for models 2, 3, and 4 using second-order autoregressive error structure.
-6
-4
-2
0
2
4
6
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
-6-4-20246
-6
-4
-2
0
2
4
6
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
-6-4-20246
-6
-4
-2
0
2
4
6
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
-6 -4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
-6-4-20246
-6
-4
-2
0
2
4
6
-6 -4 -2 0 2 4 6
Model 2
AR(2) Residuals (m)
Model 3
AR(2) Residuals (m)
Model 4
AR(2) Residuals (m)
Age-Lag1-Residuals (m) Age-Lag2-Residuals (m) Height-Lag1-Residuals (m)
Fig. 5. Relative errors (RE%) for different base ages.
1446 Can. J. For. Res. Vol. 47, 2017
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properties of height growth, i.e., polymorphism, multiple asymp-
totes, and sigmoid growth pattern.
The MAI values of height predictions calculated by model 4 are
given to check the annual height growth patterns (Fig. 7). As seen
on the graph, the MAIs of height are low for the early tree ages,
peak at the age of 50–60 years, and then decrease thereafter.
When we examined the graph, there is an inverse correlation
between the times of reaching inflection point and site index
values and a direct correlation between the MAI and site index
values. For instance, the worst site index, SI = 10 m, reached the
inflection point at an age of 60 years with an MAI of 0.11 m, while
the best site index, SI = 30 m, reached the inflection point at an age
of 51 years with an MAI of 0.34 m. However, curves for the other
two models show the direct correlation between the times of
reaching the inflection point and site index values. This growth
pattern is obtained only by model 4. As mentioned earlier, this
feature is one of the desirable biological patterns of site index
models.
Consequently, all desirable growth patterns mentioned above
were accomplished for model 4 so it was chosen as the best pre-
dictive model of site index. Model 4 (Bertalanffy–Richards), based
on the site-related parameters of a= exp(X) and c=b
2
+1/X,isthe
most appropriate model, with Radj
2of 0.9936, AIC of 119.55, and
RMSE of 0.5446, providing all height growth patterns. The final
mathematical form of the model 4 is given as follows:
h⫽exp(X0)[1 ⫺exp(⫺0.014686t)](1.227079⫹1/X0)
X0⫽1
2[(ln h0⫺1.227079L0)
⫹
兹
(ln h0⫺1.227079L0)2⫺4L0]
L0⫽ln[1 ⫺exp(⫺0.014686t0)]
4. Discussion
In this study, five different site index equations derived with
the GADA were fitted, and attempts were made to correct the
inherent autocorrelations of data. All of the models fitted based
on Lundqvist–Korf, Bertalanffy–Richards, and King–Prodan basic
growth functions showed good statistical results except models 1
and 5. Model 1 still showed the problem of autocorrelation with a
Durbin–Watson value of 2.1155. Also, this model explained about
97% of variance with high error values (AIC = 3271.40, RMSE =
1.2764). One parameter of model 5 derived from King–Prodan
growth model was not significant at the 0.05 level. However, the
other three dynamic site index models (models 2, 3, and 4) ex-
plained about 99% of the total variance in dominant height
growth, with low error values. These three GADA models also
showed no trends in residuals and ensured consistent estimates
until a height of 30 m. The predictions of trees taller than 30 m
showed lower residuals than trees from other height classes.
After statistical assessment, several desirable growth patterns
were investigated for these three dynamic site index models. All
realistic height-growth features such as polymorphism, variable
asymptotes, sigmoid growth pattern, and defined origin point at
time 0 were ensured by these models. Beside these desirable
height-growth features, the MAI value should increase with site
Fig. 6. Site index curves at a reference age of 100 years for models 2,
3, and 4.
Fig. 7. MAI values calculated for model 4.
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index values, and the age of reaching the inflection point should
decrease with site index values. This feature is also an indispens-
able growth pattern of dominant height growth, making the site
index curves more realistic. However, only model 4 derived from
the Bertalanffy–Richards growth function provided this growth
pattern and was selected as the best site index model.
Kalıpsız (1963) developed site index curves for Crimean pine by
old methodology, and these curves have been used in Turkey for
Crimean pine forests. New site index curves developed in this
study were compared with the old site curves developed by
Kalıpsız (1963) both statistically and graphically. For this purpose,
site index curves with SIs of 12, 17, 22, 27, and 32 m at the reference
age of 100 years were calculated. Results of the Wilcoxon signed-
rank test reveal that the site index curves for five site indices
developed by Kalıpsız (1963) and prepared in this study were sta-
tistically different from each other (p< 0.01). In Fig. 8, compared
with the new site index curves, site index curves for site indexes of
12, 17, 22, 27, and 32 m obtained by Kalıpsız (1963) overestimated
dominant heights until the age of 100 years and then underesti-
mated them as trees matured.
The old site index model developed by Kalıpsız (1963) for Crimean
pine stands in Turkey is both anamorphic with single asymptotes
and base age variant. In the first step of this fixed base age modeling
approach, a base age is selected to fit the equations. The estimations
of these models depend strongly on this base age being chosen at
the beginning of model fitting. In contrast to the previous site
index curves, the new curves based on GADA are completely in-
dependent of base age. The GADA models are base-age invariant
models that are independent of base age and ensure the same
dominant height estimates at any age. The site index models be-
come more flexible by the base age invariant feature.
The GADA models have been recommended in many studies
because they have more realistic height growth patterns and
more flexible estimates than earlier site index models (Cieszewski
and Bailey 2000;Cieszewski 2002,2003,2004;Diéguez-Aranda
et al. 2006;Kahriman 2011;S¸enyurt and Ercanlı 2013;Ercanlı et al.
2014). Also, many studies recommended the correction of the au-
tocorrelation error structure for site index modeling using the
stem analysis data; some of these studies are Corral-Rivas et al.
(2004),Barrio-Anta and Diéguez-Aranda (2005),Diéguez-Aranda
et al. (2005),Bravo-Oviedo et al. (2007,2008),Martín-Benito et al.
(2008),Vargas-Larreta et al. (2013),Rodríguez-Carrillo et al. (2014),
and Tewari et al. (2014).
In Turkey, site index curves developed many years ago for the
most common tree species have been used. For an accurate assess-
ment of productivity of forest areas, it is necessary to develop new
base age invariant dynamic site index models having more realis-
tic growth patterns. Also, site index models for mixed stands are
needed to evaluate the productivity of forest areas for each tree
species in the mixture. Almost all forest researchers in Turkey use
temporary sample plots and stem analysis data for the forest pro-
ductivity studies. However, there is a great need for permanent
sample plots in which to observe and evaluate the growth and
yield characteristics of the stands and trees from the first years to
matured ages.
Acknowledgement
This study was produced from a master’s thesis prepared by
Mehmet Seki and supervised by Dr. Oytun Emre Sakıcı for the
Institute of Natural and Applied Science, Kastamonu University,
Turkey. We thank Dr. I
˙lker Ercanlı for his support and Muzaffer
Büyükterzi (deceased) for his contributions. We also thank the
Associate Editor and two anonymous reviewers of the manuscript
for their valuable suggestions.
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